After replacing the linearisation with a sufficiently high tensor power (which does not change the notion of semistability), the quotient map \(q_U\)  is constructed from a linear projection \(\mathbb {P}(V) \dashrightarrow \mathbb {P}(W)\)  where \(V= H^0(Y,\mathcal {L})^*\)  and \(W = (H^0(Y,\mathcal {L})^U)^*\) . More precisely, we obtain a locally closed immersion \(Y_{\min }/U \hookrightarrow \mathbb {P}(W)\)  and we let \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}Y_{\min }/U\hspace{-0.83328pt}}\hspace{0.83328pt}\subset \mathbb {P}(W)\)  denote the projective completion. There is an induced \(L\) -action on both of these projective schemes, whose (semi)stable loci can be described in terms of the (reductive GIT) Hilbert–Mumford criterion.
Let \(T\)  be a maximal torus in \(L\)  containing \(\lambda (\mathbb {G}_m)\)  and consider the \(T\) -weights on \(V\)  and \(W\) . For \(y \in Y\)  (or strictly speaking a lift to the affine cone \(V\) ), we need to compare the convex hull of \(T\) -weights of \(y\)  and \(q_U(y)\) ; since \(q_U\)  is induced by a linear projection, it forgets some weights:
\( \operatorname{Conv}_T(y) \supset \operatorname{Conv}_T(q_U(y)). \) 
However, the sections in \(H^0(Y,\mathcal {L})\)  with maximal \(\lambda \) -weights (dually corresponding to the locus \(V_{\min } \subset V =H^0(Y,\mathcal {L})^*\)  with minimal \(\lambda \) -weights) are all \(U\) -invariant (see [1]}) and thus these weights are preserved by \(q_U\) . Let \(H_{\min } \subset \mathfrak {t}^*\)  be the hyperplane containing the \(T\) -weights on \(V\)  which are \(\lambda \) -minimal (this hyperplane is perpendicular to the ray corresponding to \(\lambda \)  when we identify \(\mathfrak {t} \cong \mathfrak {t}^*\)  using our given norm); then we have
\( \operatorname{Conv}_T(y) \cap H_{\min } = \operatorname{Conv}_T(q_U(y)) \cap H_{\min }.\) 
All the \(T\) -weights are contained in a closed half space bounded by this hyperplane and, as the linearisation is twisted to be well-adapted, the origin lies very close to this hyperplane. Points in \(Y_{\min }\)  have at least one weight on \(H_{\min }\)  and points in \(Z_{\min }\)  have all their weights on this hyperplane. The map \(p : Y_{\min } \rightarrow Z_{\min }\)  is the retraction to the minimal \(\lambda \) -weight space \(\mathbb {P}(V_{\min })\) , i.e. it throws away all the \(T\) -weights not lying on the hyperplane \(H_{\min }\) . If \(y \in Y_{\min } \setminus UZ_{\min }\) , then the weight polytope \(\operatorname{Conv}_T(q_U(y))\)  contains a weight outside of \(H_{\min }\) . If \(y\in Y^s_{\min }\) , then \(p(y) \in Z^s_{\min }\)  is \(\overline{L}\) -stable; that is, the projection of the origin to \(H_{\min }\)  is contained inside the interior of \(\operatorname{Conv}_T(y) \cap H_{\min }\) .
For the proof of (i), if \(y\in Y^s_{\min }\setminus U Z^s_{\min }\) , then we claim that the origin is contained in the interior of \(\operatorname{Conv}_T(q_U(y))\) . Indeed, the origin lies very close to \(H_{\min }\) , and since \(y\in Y^s_{\min }\)  this weight polytope meets \(H_{\min }\)  in a codimension 1 polytope \(\operatorname{Conv}_T(y) \cap H_{\min }\)  containing the projection of the origin. Since \(y \notin UZ_{\min }\) , the weight polytope also contains one weight away from \(H_{\min }\) , so it is full dimensional. Since \(Y^s_{\min }\setminus U Z^s_{\min }\)  is \(H\) -invariant, the same is true for all points in the \(H\) -orbit of \(y\) , whose image under \(q_U\)  is the \(L\) -orbit of \(y\) ; hence, by the Hilbert–Mumford criterion for the \(L\) -action on \(\mathbb {P}(W)\) , we conclude that \(q_U(y)\)  is \(L\) -stable.
For (ii), the first inclusion follows from (i), as \(\operatorname{dom}(q) = q_U^{-1}(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}Y_{\min }/U\hspace{-0.83328pt}}\hspace{0.83328pt}^{L-ss})\) . The second inclusion follows as we can pullback invariant sections along \(q_U\) . The middle equality is just the reductive Hilbert–Mumford criterion and the right inclusion holds by a Hilbert–Mumford type argument involving \(T\) -weights similar to that given in (i).
Finally (iii) follows from (i) and (ii), and the final statement follows as then \(Z_{\min }^s = Z_{\min }^{ss}\) .
